Consider an investor who chooses a portfolio to maximize his expected utility of wealth next period, where the utility function is power (CRRA) with coefficient of relative risk aversion γ . Assume that there are two assets, a riskless asset with log return rf and a risky asset with log return r. Assume that the log risky return is normal with variance σ2 and that the optimal portfolio places a weight α on the risky asset. Use the approximation to the log portfolio return rp,t+1 – rf ,t+1 =log(1 + αt(exp(rt+1 – rf ,t+1) – 1))≈ αt(rt+1 – rf ,t+1) +1/2*αt(1 – αt)σ2t , to analyze this model. Also use the formulation of the Sharpe ratio in terms of moments of log returns that is, write the Sharpe ratio as S = (E[r ] – rf + σ2/2)/σ.

(a) Derive an expression for the value function (the investor’s maximized utility) as a function of initial wealth, the riskless interest rate, the coefficient of relative risk aversion, and the Sharpe ratio of the risky asset.
(b) Suppose that the time period is one year. The log riskless interest rate is 2%, the log of the expected simple return on the risky asset is 10%, and the standard deviation of the log risky asset return is 40%. The investor initially chooses to hold 25% of his portfolio in the risky asset. What must his risk aversion be?
(c) Now suppose that the standard deviation of the log risky asset return declines to 20%, but the riskless interest rate and the expected simple return on the risky asset do not change. (This could result from improved diversification of underlying idiosyncratic risky opportunities that are packaged into a single fund that is marketed to the investor.) If the investor’s portfolio weight in the risky asset does not change, show that the effect on the value function is equivalent to an
increase in the riskless interest rate of a percentage points. What is a? If the investor adjusts his portfolio weight in the risky asset to its new optimal level, show that the effect on the value function is equivalent to an increase in the risk-less interest rate of b percentage points. What is b? Is b larger or smaller than a?Explain.